Page:American Journal of Mathematics Vol. 2 (1879).pdf/20

14 It is assumed, and assumed only in the "Common Theory of Flexure" put forth by Mariotte and Leibnitz, that the intensities of the normal internal stresses parallel to the neutral surface vary directly as the first powers of the normal distances from that neutral surface. This assumption gives results corresponding to experimental ones, with degrees of approximation varying according to the nature of the material and the shape of the piece subjected to bending. Its chief merit, and a very great one, is that it leads to very simple discussions of the eases which ordinarily occur in practice. It ignores, however, the existence of any internal shearing stress, and the formulae deduced for deflection do not involve the distortion which any piece of material suffers when subjected to the action of external forces.

Nevertheless, the method of fixing the position of the neutral surface is correct, since it is based on one of the first principles of statics, i. e., that each of the sums of the components of the internal stresses, taken along three rectangular axes, must be equal to zero. The sum of the component forces of each sign along any axis, and not the sum of the component moments, must be equal to each other when the external forces act in a direction normal to the axis of the beam.

Navier first assumed the equality of the moments, but soon after abandoned the idea and pronounced it erroneous.

The principle just stated, first given by Parent, will be used in the following discussion in the determination of the position of the neutral surface.

Two assumptions will be made, the last only of which, however, as will eventually be shown, tends to give the investigation and approximate character.

The one source of approximation which probably causes the discrepancy between the results which follow and those of experiments is the neglect of lateral contraction and expansion; and those phenomena will be noticed further on.

It will first be assumed that the material has a non-crystalline structure. This is not absolutely necessary, but it emphasizes the proof that the results apply to material of any kind.

The second assumption is this, that the applied bending forces produce no compression at their points of application. This really amounts to supposing the bending to be produced by a single force acting at the proper distance from the section under consideration, while the portion of the beam on the other side of the section is held in position by the requisite forces.